Question

At a party each guest shook hands with every other guest exactly once. There were a total of 105 handshakes. How many guest were there?

Answer 1

There are `15` guests

Explanation:

If there are `n` guests then each guest shakes hands with `(n-1)` other guests. Note, however, that each time a handshake occurs it counts as `2` handshakes (one for each person involved).

Therefore with `n` guests the number of handshakes will be `(nxx(n-1))/2`

We are told
`color(white)("XXX")(nxx(n-1))/2=105`

Therefore
`color(white)("XXX")n^2-n=210`

`color(white)("XXX")n^2-n-210=0`

`color(white)("XXX")(n+14)(n-15)=0`

Therefore
`color(white)("XXX")n=-14` or `n=15`
And since the number of guest can not be negative:
`color(white)("XXX")n=15`